Credit cards, statements and fees

Credit card interest compounded daily, reading a monthly statement, minimum repayments and why paying only the minimum takes so long, and the fees charged on credit cards.

In Year 11 you met the interest-free period and simple daily interest on a credit card. Most cards actually compound interest daily: each day's interest is added to the balance and charged interest itself. Combine that with rates near $20\%$ p.a. and a small minimum repayment, and a modest debt can follow you for decades.

Interest compounded daily

Credit card rates are quoted per year. For daily compounding, the rate per day is the annual rate divided by $365$, and the number of periods is the number of days.

Credit card interest compounded daily

$$I = P\left(1 + \frac{r}{365}\right)^{d} - P$$

Here $P$ is the balance being charged interest, $r$ is the annual rate as a decimal and $d$ is the number of days.

Example 1●●

Ella has an unpaid balance of $\$2400$ on a card charging $19.99\%$ p.a. (an example rate), compounded daily. Find the interest for a $31$-day month.

$\dfrac{r}{365} = \dfrac{0.1999}{365}$, $d = 31$

Daily rate and number of days.

$I = 2400\left(1 + \dfrac{0.1999}{365}\right)^{31} - 2400 \approx \$41.08$

Find the amount owing, then subtract the balance.

Simple interest would be $2400 \times \dfrac{0.1999}{365} \times 31 \approx \$40.75$. Compounding adds a little each month, and it builds up over time.

Compare with simple daily interest from Year 11.

Reading a statement

Each month the bank sends a statement. The key line is the closing balance:

Closing balance

$$\text{closing} = \text{opening} + \text{purchases} + \text{interest} + \text{fees} - \text{payments}$$

The statement also shows the minimum repayment, usually a small percentage of the closing balance (for example $2\%$) or a fixed amount such as $\$25$, whichever is greater, and the due date. Paying the full closing balance by the due date usually means no interest on new purchases. Paying less means interest on everything.

Example 2●●

A statement shows an opening balance of $\$1250$, purchases of $\$640$, a payment of $\$300$, interest of $\$22.15$ and a late payment fee of $\$15$. The minimum repayment is $2\%$ of the closing balance or $\$25$, whichever is greater. Find the closing balance and the minimum repayment.

Closing $= 1250 + 640 + 22.15 + 15 - 300 = \$1627.15$

Add what increases the debt, subtract what reduces it.

$2\% \times 1627.15 = \$32.54$, which is more than $\$25$

Find 2% and compare it with the 25-dollar floor.

Minimum repayment $= \$32.54$

The larger amount is the minimum.

The minimum repayment trap

At around $20\%$ p.a., one month's interest is about $1.7\%$ of the balance. A $2\%$ minimum repayment therefore pays off only about $0.3\%$ of the balance each month. The rest of the payment is just interest.

Example 3●●●

Aiden owes $\$3000$ on a card at $20.99\%$ p.a., compounded daily, and makes no new purchases. Compare paying the minimum ($2\%$ or $\$25$) with paying a fixed $\$150$ a month.

Minimum only: about $445$ months (over $37$ years), with about $\$13\,175$ in interest.

A spreadsheet or the simulator runs the balance month by month.

Fixed $\$150$: $25$ months, with about $\$733$ in interest.

A fixed payment is far bigger than the interest, so the balance falls fast.

Paying $\$150$ a month saves about $\$12\,400$ and 35 years.

Compare.

InteractiveMinimum payment trap

Try this

Set $\$3000$ at $20.99\%$ and pay the minimum. Check Example 3. Now switch to a fixed payment and find the smallest amount that clears the debt within $3$ years.

Fees

Fees add to the balance and are then charged interest like any purchase. Typical fees (amounts vary by card):

Annual fee, often $\$0$ to a few hundred dollars for rewards cards.
Late payment fee if the minimum is not paid by the due date.
Cash advance fee for withdrawing cash, often about $3\%$ of the amount. Cash advances also have a higher rate and no interest-free period: interest starts on the day.
Foreign transaction fee, often about $3\%$ of purchases made in another currency or from an overseas website.

Example 4●●

Ruby withdraws $\$400$ cash on her card. There is a $3\%$ cash advance fee, and cash advances are charged $21.99\%$ p.a. compounded daily from the day of withdrawal. She repays it $20$ days later. Find the fee and the interest on the $\$400$.

Fee $= 0.03 \times 400 = \$12$

Fee is 3% of the amount withdrawn.

$I = 400\left(1 + \dfrac{0.2199}{365}\right)^{20} - 400 \approx \$4.85$

Interest compounds daily for 20 days, with no interest-free period.

$12 + 4.85 = \$16.85$, which is about $4.2\%$ of $\$400$ in under three weeks.

Total cost of borrowing 400 dollars for 20 days.

Your turn

A card balance of $\$3200$ is charged $18.25\%$ p.a. interest compounded daily for $45$ days. Find the interest.

Show the working

Daily rate $= 0.1825 \div 365 = 0.0005$. $I = 3200(1.0005)^{45} - 3200 \approx \$72.80$.

Flashcards

Credit card interest compounded daily

$I = P\left(1 + \dfrac{r}{365}\right)^{d} - P$

Closing balance

Opening $+$ purchases $+$ interest $+$ fees $-$ payments

Why the minimum repayment is a trap

Most of it is interest, so the balance barely falls and the debt lasts for decades.

Practice

A credit card balance of $\$1500$ is charged $21.9\%$ p.a. interest compounded daily. Find the interest for $30$ days, to the nearest cent.
The minimum repayment is $2\%$ of the closing balance or $\$25$, whichever is greater. Find the minimum on a closing balance of $\$960$.
Using the same rule, find the minimum repayment on a closing balance of $\$1840$.
A statement shows an opening balance of $\$1250$, purchases of $\$640$, interest of $\$22.15$, a late fee of $\$15$ and a payment of $\$300$. Find the closing balance.
Why does paying only the minimum take so long to clear a credit card debt?
Leo buys headphones from an overseas website. The purchase converts to $\$380$ and the card charges a $3\%$ foreign transaction fee. Find the fee.
A cash advance of $\$400$ is charged $21.99\%$ p.a. interest compounded daily from the day it is taken. Find the interest after $20$ days, to the nearest cent.
A card charges $18.25\%$ p.a. Find the daily interest rate, as a percentage.
Exam style: A credit card balance of $\$3200$ is charged $18.25\%$ p.a. interest compounded daily. Find the interest charged over $45$ days, to the nearest cent.
Exam style: A closing balance is $\$2450$ and Sophie pays only the minimum of $2\%$ or $\$30$, whichever is greater. The next month's interest is $1.7\%$ of the remaining balance, and she makes no purchases. Find the next closing balance, to the nearest cent.